Exact formulas in some boundary crossing problemsfor integer-valued random walks

For a wide class of integer-valued random walks,we obtain exact expressions for the distributionof the first excess over leveland the corresponding renewal function as well as for the distribution of the trajectory supremumif it is finite.We discuss possibilities of obtaining explicit expressionsfor pre-stationary and stationary distributionsof a random walk with switchings at the strip boundaries.The research is based on the factorization representationsfor the double moment generating functions of the distributions under study.

integer-valued random walk, factorization method, excess over boundary, renewal function, oscillating random walk